Chapter 11Length and Area

11.1 Areas of Triangles and Parallelograms•Area of a square = s2

•Area of a rectangle = bh•Area of a parallelogram = bh•Area of a triangle = bh

2

Important Information

•If two polygons are congruent, then they have the same area.

•The area of a region is the sum of the areas of its non-overlapping parts.

11.2 Areas of Trapezoids, Rhombuses, and Kites

Formulas

•Area of a trapezoid = h(b1 + b2) 2

•Area of a rhombus = d1 d2

2

More Formulas

•Area of a kite = d1 d2

2

Examples•Find the area of RSTU.•Find the area of the polygon

RSTWUV.

28

35

2520

W

VU T

SR

Example

•One diagonal of a kite is 1/3 as long as the other. The area of the kite is 0.24m2. What are the lengths of the diagonals?

11.3 Perimeter and Area of Similar Figures

•If two polygons are similar with the lengths of corresponding sides in the ratio a:b, then the ratio of their areas is a2:b2.

Example

•ABCD is similar to RSTU.•Find the ratio of the perimeters.•Find the ratio of the areas.

1018

B C

A D

S

R U

T

Example

You are painting 2 wall in an office complex that are similar in shape- both rectangles. One wall has a side length of 10ft. The corresponding side of the other wall is 14ft. You need 7 quarts to paint the larger wall. How many quart do you need for the smaller wall?

Example

•The Pentagon in Washington DC is a regular pentagon with side lengths of 900ft. The area is 1,400,000ft2. The perimeter of a scale model of the building is 30yds. What is the area of the scale model?

11.4 Circumference and Arc length•Circumference – the distance around the

circle

C = 2r = dr

d

•Arc length- a portion of the circumference of a circle

= mAB

360arc length AB

2r

r

P B

A

Find the length of AB

•1. 2.

45

9in.

P B

A

90

15in.

P B

A

Find the indicated measures.

• Find the circumference.

756.82cm

T

Find the mAB.

r = 15.28 ft.

AB = 40ft.

B

A

11.5 Area of Circles and Sectors•Area of a circle A = πr2

•Sector of a circle- region bounded by 2 radii of the circle and the intercepted arc.

•Sector APB = m AB πr2 360o

r

P

A

B

Example

•Find the area of the sectors formed by <HJK.

100

12cm

J

H

K

Example

•Find the area between the large outer circle and the two smaller circles.

3m

11.6 Areas of Regular Polygons

•The center of a polygon and the radius of a polygon are the center and radius of its circumscribed circle.

•Apothem of a polygon- distance from center to any side of a polygon

•Central Angle of a regular polygon- the angle formed by 2 radii drawn to consecutive vertices of the polygon

Vocabulary

•Apothem

•Radius

•Central AngleN

M

P

Q

Formula

•Area of a regular polygon A =

N

M

P

Q

Examples

•For a regular octagon inscribed in circle C, find the following:

•m<RCY =

•m<RCZ =

•M<ZYC =

C

R

Y

Z

Example

•What is the area of a regular hexagon with a side length of 8 inches?

Example

•What is the area of a regular decagon inscribed in a circle with a radius of 8mm?

11.7 Using Geometric Probability•Probability (of an event) - P(A)- a

measure of the likelihood that an event will occur.

•Geometric probability is a ratio that involves a geometric measure like length or area.

Example•Find the probability that a point chosen at

random on AE is on CD.

-4 -3 -2 -1 0 1 2 3 4 5

A B C D E

Example

•A dart game uses targets with concentric circles of radii 5,8, and 12 inches. What is the probability the dart will earn 20 points?

20

5

15